‎‎In this article‎, ‎for a lattice $\mathcal L$‎, ‎we define and investigate‎ ‎the annihilator graph $\mathfrak {ag} (\mathcal L)$ of $\mathcal L$ which contains the zero-divisor graph of $\mathcal L$ as a subgraph‎. ‎Also‎, ‎for a 0-distributive lattice $\mathcal L$‎, ‎we study some properties of this graph such as regularity‎, ‎connectedness‎, ‎the diameter‎, ‎the girth and its domination number‎. ‎Moreover‎, ‎for a distributive lattice $\mathcal L$ with $Z(\mathcal L)\neq\lbrace 0\rbrace$‎, ‎we show that $\mathfrak {ag} (\mathcal L) = \Gamma(\mathcal L)$ if and only if $\mathcal L$ has exactly two minimal prime ideals‎. ‎Among other things‎, ‎we consider the annihilator graph $\mathfrak {ag} (\mathcal L)$ of the lattice $\mathcal L=(\mathcal D(n),|)$ containing all positive divisors of a non-prime natural number $n$ and we compute some invariants such as the domination number‎, ‎the clique number and the chromatic number of this graph‎. ‎Also‎, ‎for this lattice we investigate some special cases in which $\mathfrak {ag} (\mathcal D(n))$ or $\Gamma(\mathcal D(n))$ are planar‎, ‎Eulerian or Hamiltonian.